Complex Analysis

The theory of functions of a complex variable is a central theme in mathematical
analysis that has links to several branches of mathematics. Understanding
the basics of the theory is necessary for anyone who wants to have a general
mathematical training or for anyone who wants to use mathematics in applied
sciences or technology.

The book presents the basic theory of analytic functions of a complex variable
and their points of contact with other parts of mathematical analysis. This results
in some new approaches to a number of topics when compared to the current
literature on the subject.

Some issues covered are: a real version of the Cauchy–Goursat theorem, theorems
of vector analysis with weak regularity assumptions, an approach to the
concept of holomorphic functions of real variables, Green’s formula with multiplicities,
Cauchy’s theorem for locally exact forms, a study in parallel of Poisson’s
equation and the inhomogeneous Cauchy–Riemann equations, the relationship
between Green’s function and conformal mapping, the connection between
the solution of Poisson’s equation and zeros of holomorphic functions, and the
Whittaker–Shannon theorem of information theory.

The text can be used as a manual for complex variable courses of various levels
and as a reference book. The only prerequisites for reading it is a working knowledge
of the topology of the plane and the differential calculus for functions of
several real variables. A detailed treatment of harmonic functions also makes the
book useful as an introduction to potential theory.